Friday, July 26, 2013

Miss Liberty 5: Outnumbering #2

Miss Liberty and I were standing at
the bottom of a small amphitheater. A little way up the slope was a control
panel, and standing before it was a black sphere six feet in diameter.

Miss Liberty pointed out a slogan
written on the side of the control panel: ON COMPANY BUSINESS. We nodded grimly
at each other; for that could mean only one thing. It meant that we were in the
clutches of the Celestial Intervention Agency; the deadliest, trickiest, and
most bureaucratic spy agency in the Universe!

With a loud WHIRR the black sphere
turned on its vertical axis, revealing an opening; it was really a large round
chair. A man sat inside. He was dressed in a sober business suit; he was rather
corpulent; he wore sideburns and a mustache, though the top of his head was
bald.

“Who are you?” asked Liberty.

He said, “I am #2.”

Liberty asked, “Who is #1?”

#2 said scornfully, “You are #0.”

“I am not a number,” said Liberty.
“I am a Free Spirit!”

“You are here for information,” said
#2. “I shall provide you with that information, then hand you over to my agents
for final disposal. State three questions.”

“Well, I was wondering about this
map,” said Miss Liberty, taking a badly-worn chart out of her robe. “Look: it’s
all divided up into countries. I don’t see why, by the way; people are much the
same anywhere you go.”

“And what about this map?” #2 asked
warily.

“It needs some colors,” she
complained. “As you can see, it’s all in black and white. A line sketch; how
drab! It would be so nice to color these areas in.”

“Obviously, no two neighboring
countries should be the same color,” #2 said, “or we couldn’t tell the
boundary.”

“And there’d be no point to a
boundary,” Liberty added. “So what I was wondering is this; how many colors
suffice to color this map?”

“Four colors suffice for all planar
maps,” #2 said bluntly.

“You’re sure,” Liberty said.

“Quite sure. My operatives have
proven it,” said #2. He pushed a button on the control console; a panel opened
on the side facing us; and an immensely thick computer printout slid out. It
fell with a loud THUD at our feet.

“That computer printout is a formal
proof of the Four Color Map Theorem,” said #2. “Examine it at your leisure.”

Miss Liberty and I looked at each
other; then we shrugged our shoulders. Maybe that was a proof. I suppose
it was, if their theoretical analysis was right, their computer was sound, and
its programs were correct. I don’t know; what do you think?

“Ask your next question,” said #2.

“You people like to be efficient; so
I’m sure you can help me with this next problem,” said Miss Liberty. “You see,
I want to work out a minimum-distance tour.”

“So you’re going on a tour, are
you?” #2 asked Miss Liberty. He appeared interested in this information.

Miss Liberty said, “You see, one of
these years, sooner than you expect, I’m going to take a tour of the 365
largest cities on Planet Earth. I’ll visit one city per day, visiting each city
once.”

#2 said, “Quite a year!”

Liberty turned her map over; on the
other side was a large number table. “Here’s a list ofthose cities, and the distances between them.
Which tour is shortest?”

#2 frowned. He pushed a button on
the control panel. It flickered its lights,beeped, and issued a printed paper tape. #2 read it and said, “That is
difficult to determine. You have assigned us the Traveling Salesman Problem for
365 nodes. The total number of possible tours is 2.5 times 10778.
Checking all of these routes for the shortest one would be time-consuming, to
say the least.”

“Then you had better get started
right away.”

“We would prefer a more efficient
algorithm.”

Liberty asked, “Do you have one?”

#2 said, “Not as yet.”

“Maybe there isn’t any.”

“That possibility too has not yet
been ruled out.” #2 sighed, then pushed a button on the panel. “A portion of
the system has been assigned to the task of checking all the routes. It will
report back to us as soon as its findings are complete. Next question, please,”
#2 said irritably.

“You don’t have the best route yet?
Too bad,” saidLiberty. “Would you like
to know mine?”

“Yes, we would very much indeed like
to know what route you plan to take,” said#2.

“Here it is!” she said. And she
produced a thin computer printout which she dropped lightly at #2's feet.
“That’s a pair of large numbers,” she explained. “The first number encodes the
route, and the second one is the encryption key.”

“Then this shouldn’t be hard to
decode,” #2 said as he bent down to pick up the printout.

“It shouldn’t be,” said Miss
Liberty, “if you had the decryption key-number. But you only have the encryption
key there. You can only make code, not break it.”

#2 held the thin printout and said
with dismay. “A dual-key public encryption scheme.”

“So now you can write me any
messages you want to, and be sure that none of those nasty little spies are
eavesdropping!” said Miss Liberty. “Isn’t that a relief?”

“How do we decode this?” #2
demanded.

“That’s easy. To crack the code, you
just have to factor the second number. It’s a product of two large prime
numbers. They’re only twenty thousand digits long,” said Miss Liberty.

#2 roared, “Twenty thousand digits!”

“It’s a simple program; it’ll just
take awhile, that’s all!”

“The other program will finish
sooner!”

“Don’t you have any factoring
programs fast enough?”

“No.” #2 glared at Miss Liberty;
then he sighed with resignation. He fed the thin printout into the control
panel and pushed a button.

Then the lights went out.

In the sudden darkness we heard #2
say, “This program is jammed. Consult supervisor.”

* click *

mmmmmmmmmmmmmmmmMMMMMMMMMMMMM....

Thus Miss Liberty outnumbered #2
with the Traveling Salesman Problem and a large number factorization.

#2 said, “You now have Full Access
to JOV. Beware, Master! She’s a glitch! Burn her!”